Let \( u = x^2 + 2 \). Then, \( h(u) = 3\sqrt{u - 2} + 5 \). We want to find \( h(x^2 - 2) \), so set \( u = x^2 - 2 \). Then,
["Understanding the Function Composition: Evaluating ( h(x^2 - 2) )", "In advanced mathematics, function composition allows us to build complex behaviors by combining simpler functions. Consider two functions defined as:", "Let\n[\nu = x^2 + 2,\n]\nand\n[\nh(u) = 3\sqrt{u - 2} + 5.\n]", "Our goal is to find the expression for ( h(x^2 - 2) ). To do this efficiently, we follow a clear process using substitution.", "### Step 1: Understand the Role of ( u )", "The function ( h ) is defined in terms of ( u ), not ( x ). This means whenever we compute ( h ) at some input, we first replace ( x ) with ( u ). So, to evaluate ( h(x^2 - 2) ), we must treat ( u = x^2 - 2 ), just as before we used ( u = x^2 + 2 ) to compute ( h(u) ).", "### Step 2: Substitute ( u = x^2 - 2 ) into ( h(u) )", "We substitute ( u = x^2 - 2 ) directly into the expression for ( h(u) ):", "[\nh(x^2 - 2) = 3\sqrt{(x^2 - 2) - 2} + 5\n]", "Simplify the expression inside the square root:", "[\nh(x^2 - 2) = 3\sqrt{x^2 - 4} + 5\n]", "### Step 3: Final Result", "Thus, the composition of ( h ) with ( x^2 - 2 ) yields:", "[\nh(x^2 - 2) = 3\sqrt{x^2 - 4} + 5\n]", "This result is valid for all real ( x ) such that ( x^2 - 4 \geq 0 ), meaning ( |x| \geq 2 ), since the square root requires a non-negative radicand.", "---", "This example illustrates the power of function composition: once we define how ( h ) behaves on ( u ), we can dynamically evaluate it for different input forms by substituting appropriately. Whether working in calculus, modeling physical systems, or computer science algorithms, mastering this technique enables elegant and efficient problem-solving.", "Explore how redefining input variables empowers deeper analysis—an essential skill in both theoretical and applied mathematics."]
